It is important to point out that formulas of the type (REF ) have already been investigated in the literature in contexts not necessarily related to the QHE. For instance, in [1]} a similar formula has been derived for the case of a (\(d\) -dimensional) noncommutative torus.
The latter result provides a direct generalization of the first Connes'
formula obtained in [2]} for the discrete
case. Even more related to Theorem REF  is the work [3]}. In the latter paper a formula of the type (REF ) is derived for the noncommutative Euclidean space (or Moyal plane) which is indeed similar to the algebra \({C}_B\)  introduced above. The main difference between the result in [3]} and Theorem REF  lies in the choice of the Dirac operator. In fact, in [3]} the authors deal with the operator \(\widehat{D}\)  (described above) which has a non-compact resolvent, while formula (REF ) is obtained from the Dirac operator \(D_B\)  which has compact resolvent.
